Let , be real and z be a complex number.

Question

Let $\alpha,\beta$ be real and $z$ be a complex number. If $z^2+\alpha z+\beta=0$ has two distinct roots on the line $\operatorname{Re}(z)=1$, then it is necessary that:

Accepted Answer

As real part of roots is $1$

Let roots are $1 + pi,1 + q$

$$\therefore$$ sum of roots $$ = 1 + pi + 1 + qi =  - \alpha $$

which is real $$ \Rightarrow q =  - p\,\,$$

or root are $1+pi$ and $1-pi$

product of roots $$ = 1 + {p^2} = \beta  \in \left( {1,\infty } \right)$$

$$p \ne 0$$ as roots are distinct.

Solutions · 1